4e5n Amath P1 Prelim 2009 With Ans

Name: _______________________________ (

)

Class: _______

MONTFORT SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009

Secondary 4 Express / 5 Normal Academic

ADDITIONAL MATHEMATICS PAPER 1 Thursday

17 September 2009

4038/01 1030 – 1230

2 hours

INSTRUCTIONS TO CANDIDATES Write your name, register number and class on the question paper and writing paper provided. Attempt all questions. Write your answers and working on the separate writing paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in case of angles in degrees, unless a different level of accuracy is specified in the question INFORMATION FOR CANDIDATES The number of marks is given in brackets [ question.

] at the end of each question of part

The total number of marks for this paper is 80. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers.

This document consists of 5 printed pages including the cover page. Setter: Tan TB

[Turn Over

1

Mathematical Formulae 1.

ALGEBRA

Quadratic Equation For the equation

ax 2 + bx + c = 0 ,

x=

− b ± b 2 − 4ac 2a

Binomial Theorem

n n n = a n +  a n−1b +  a n − 2 b 2 + ... +  a n − r b r + ... + b n , 1  2 r n n! where n is a positive integer and   = .  r  (n − r )!r!

(a + b )n

2.

TRIGONOMETRY

Identities

sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cos ec 2 A = 1 + cot 2 A sin ( A ± B ) = sin A cos B ± cos A sin B cos ( A ± B ) = cos A cos B m sin A sin B tan A ± tan B tan ( A ± B ) = 1 m tan A tan B sin 2 A = 2 sin A cos A cos 2 A = cos 2 A − sin 2 A = 2 cos 2 A − 1 = 1 − 2 sin 2 A 2 tan A tan 2 A = 1 − tan 2 A 1 1 sin A + sin B = 2 sin ( A + B ) cos ( A − B ) 2 2 1 1 sin A − sin B = 2 cos ( A + B ) sin ( A − B) 2 2 1 1 cos A + cos B = 2 cos ( A + B ) cos ( A − B ) 2 2 1 1 cos A − cos B = −2 sin ( A + B ) sin ( A − B ) 2 2 Formulae for

∆ABC a b c = = . sin A sin B sin C a 2 = b 2 + c 2 − 2bc cos A. ∆=

1 bc sin A. 2

2

1

2

 −2 3  −1 Given that A =   , find A and hence solve the simultaneous equations 1 − 5   7 −2 x + 3 y − = 0, 15 1 x − 5 y + 2 = 0. 3

4   The perimeter of a rectangle is  2 3 + + 6 5  cm. Given that its length is 5  

(

)

3 + 2 5 cm, find, without using calculator, the area of the rectangle

in surd form.

3

(a)

(b)

4

5

[6]

Find all the angles between 0° and 360° which satisfy the equation 14 sin x − 3cos 2 x = 9 . Find all the angles between 0 and 5 radians which satisfy the equation π  cos  x +  = 5cos x . 2 

[4]

[4]

[4]

(i)

Find the range of values of k for which the expression 2 x 2 − kx − 3 x + k + 3 is never negative for all real values of x. [4]

(ii)

Hence, state the values of k when 2 x 2 − kx − 3 x + k + 3 = 0 has only repeated roots. Calculate the value of x for each value of k. [3]

The polynomial P ( x) = 3 x 4 + ax3 − 3bx 2 + 2a has a factor x 2 + x − 2 . Find, (i)

the value of a and of b,

[4]

(ii)

the other quadratic factor of P(x).

[2]

3

6

7

Differentiate with respect to x (a)

( 3x

(b)

ln

(a)

Write down, in descending powers of x, the first three terms in the expansions of

(b)

− 1) 2 x 2 + 1

[3]

5+ x 5− x

[3]

2

(i)

(1 − px)8 , where p is a constant.

[1]

(ii)

(1 − x )

[1]

n

, where n is a positive integer.

Given that the first three terms in the expansion of (1 − px)8 (1 − x)n are 1 − 6 x + 16 x 2 . Find the value of p and of n.

8

(a)

(b)

9

Given that the equation of a circle is ( x + 3) + ( y − 4 ) = 9 , determine whether the point P(–2, 2) is inside, outside or on the circle.

[4]

A circle whose centre lies on the line 5 x + 3 y = 0 , passes through the points (–3, 0) and (0, 1). Find the equation of the circle.

[5]

2

A curve has the equation (i)

[6]

( x − 1) y= x−2

2

3

, x ≠ 2.

Find the equation of the normal to the curve at the point where the curve crosses the y-axis.

[4]

(ii)

Find the x-coordinates of the two stationary points of the curve.

[2]

(iii)

By considering the sign of stationary points.

dy , or otherwise, determine the nature of the dx [2]

4

10

In two concentric circles, the radius of the larger circle, r1 , is five times the radius of the smaller circle, r2 . 24 2 (i) Show that the area between the two circles is π r1 . [2] 25 (ii)

11

Given that the radius of the bigger circle increases at a rate of 4 cm/s. Find the rate of increase of the area between the two circles at the instant when the radius of the smaller circle is 2 cm. Leave your answer in terms of π. [4]

Answer the whole of Question 11 on a piece of graph paper. The table below shows the experimental values of two variables x and y. x y

1.0 3.2

1.5 5.2

2.0 7.1

2.5 9.2

3.0 11.2

3.5 13.3

It is known that x and y are related by the equation y = hx + k x , where h and k are constants. y (i) Draw the graph of against x for the given data. x (ii) Use your graph to find the value of h and of k. (iii)

Estimate the value of x when y = 5 x .

(iv)

By drawing a suitable straight line, estimate the values of x and y which satisfy the simultaneous equations y = hx + k x,

y = −4 x + 9 x

[4] [3] [2]

[3]

END OF PAPER

5

Montfort Secondary School Sec 4E Additional Mathematics P1 2009 Answer key

1

 5 − 7 −1 A = − 1   7

2

Breath =

3

(a) (b) (i) (ii) (i) (ii)

4 5 6

(a)

3 −  7 , 2 −  7

2 x= , 3

y=

7 5 2 or 5+ , 5 5 41.8°, 138.2° 1.77 rad, 4.91 rad −3 ≤ k ≤ 5 k = -3 & 5, x=0&2 a = 2, b = 3 3x 2 − x − 2 18 x3 + 4 x

3 5 Area = 14 +

7 15 5

7

(a)

8

(b) (a)

2x2 + 1 1 1 1  +   2 5+ x 5− x  n ( n − 1) 2 1 − nx + x + ....... (i) 2  n ( n − 1)  2 (ii) 1 − ( 8 p + n ) x +  28 p 2 +8np +  x + ...... 2   n =2, p = ½ pt P is inside the circle

(b)

( x + 3) + ( y − 5 )

(b)

9

(a)

10

(b) (c) (ii)

11

(i) (ii) (iii) (v)

2

2

= 25

4 1 x+ 5 2 x = 1, 2½ pt of inflexion & min pt 76.8π y =h x +k x h = 4.56, k = -1.4 x = 1.96 y Suitable line: = 9 x − 4 , x = 0.3364, y = 0.754 x y=

6